between the category \(\infty \, \mathrm {Sh}(B)\)  of simplicial presheaves on \(B\)  and
the category \({K}/B\)  of spaces over \(B\) . Moreover (see [1]}) this adjunction extends to
a Quillen equivalence when \(\infty \, \mathrm {Sh}(B)\)  is equipped with a certain model structure
(the fibrant objects in \(\infty \, \mathrm {Sh}(B)\)  are the so-called \(\infty \) -sheaves or homotopy sheaves on \(B\) ).
From this perspective the model structure \(({K}/B)_{ij}\)  is perhaps more interesting
than the \(f\) -model structure on \({K}/B\) . The \(f\) -model structure and the \(ij\) -model structure
on \({K}/B\)  are related however — in [2]} it is proven that the identity functor
on \({K}/B\)  is left Quillen from the \(ij\) -model structure to the \(f\) -model structure.
